Problem: Given that $-4\leq x\leq-2$ and $2\leq y\leq4$, what is the largest possible value of $\frac{x+y}{x}$?
Solution: We can write
\[\frac{x + y}{x} = 1 + \frac{y}{x}.\]Note that $x$ is always negative and $y$ is always positive.  Thus, to maximize $\frac{y}{x},$ we should take the smallest value of $x$ and the smallest value of $y,$ which gives us
\[1 + \frac{2}{-4} = 1 - \frac{1}{2} = \boxed{\frac{1}{2}}.\]